- Expand the numerator \((3m^{-2}n)^{-3}\) to \(\frac{1}{27} m^6 n^{-3}\) using exponent rules.
- Divide the coefficients \(\frac{1}{27}\) by \(6\) to get \(\frac{1}{162}\).
- Simplify the variable terms \(m^6/m^1\) and \(n^{-3}/n^{-2}\) to obtain \(m^5\) and \(n^{-1}\).
- The final simplified expression is \(\boxed{\frac{m^5}{162 n}}\).
Explanation
- Simplifying the Numerator
To simplify the given expression \(\frac{(3m^{-2}n)^{-3}}{6mn^{-2}}\), we first analyze the numerator and denominator separately. The numerator is \((3m^{-2}n)^{-3}\). According to the power of a product rule, \((abc)^k = a^k b^k c^k\), we can distribute the exponent of \(-3\) to each factor:
\[(3m^{-2}n)^{-3} = 3^{-3} \cdot (m^{-2})^{-3} \cdot n^{-3}\]
Since \(3^{-3} = \frac{1}{3^3} = \frac{1}{27}\) and \((m^{-2})^{-3} = m^{(-2) \times (-3)} = m^6\), the numerator simplifies to:
\[\frac{1}{27} m^6 n^{-3}\]
- Simplifying the Entire Fraction
Now we combine this with the denominator, which is \(6mn^{-2}\). The full expression is:
\[\frac{\frac{1}{27} m^6 n^{-3}}{6 m n^{-2}}\]
We can separate the constants and the variables:
\[\left( \frac{1/27}{6} \right) \cdot \left( \frac{m^6}{m^1} \right) \cdot \left( \frac{n^{-3}}{n^{-2}} \right)\]
For the constants: \(\frac{1}{27} \div 6 = \frac{1}{27 \times 6} = \frac{1}{162}\).
For the variable \(m\): \(\frac{m^6}{m^1} = m^{6-1} = m^5\).
For the variable \(n\): \(\frac{n^{-3}}{n^{-2}} = n^{-3 - (-2)} = n^{-3 + 2} = n^{-1} = \frac{1}{n}\).
- Final Conclusion
Combining all the simplified parts, we get:
\[\frac{1}{162} \cdot m^5 \cdot \frac{1}{n} = \frac{m^5}{162n}\]
This result matches the first option provided. Thus, the simplified form of the expression is \(\frac{m^5}{162 n}\).
Examples
In scientific notation and engineering calculations, expressions often involve products and powers of variables (like physical constants or dimensions). Simplifying such expressions allows engineers to work with more manageable formulas, reducing the risk of errors in complex simulations or structural designs. For example, when calculating the stress on a beam, if the variables representing dimensions are grouped through simplification, the resulting formula becomes much easier to compute.